Abstract

The aim of this paper is to obtain coefficient estimates of the integral operator of the form: and using the relationship between starlike and convex functions and give its implication to disease control. Also, we obtain the growth and distortion theorems for the operator.

1. Introduction and Preliminaries

Let A denote the class of normalized univalent functions of the form

(1)

which are analytic in the unit disc with

In the class of analytic functions in (1) above, the coefficients is the quotient of the nth derivative of the function been expressed in Taylor series expansion and n!. Derivatives, whether partial or total, describe the slope of a function which is the steepness of a line. It is a known fact that the larger the slope, the steeper the line and vice versa. The coefficients play prominent role in analytic functions. Thus, in applying analytic functions to any area of life, one needs to consider it's coefficient bounds. The author in ⁶ stated that starlike and convex functions, which are aspect of analytic functions have its applications in human physiology, physical and natural phenomenon. Analytic functions can also be applied to curtailing the spread of any disease, or its occurrence. Here, we think of applying analytic functions to prevention of population of carrier of disease organism from being transformed to infectious population and thereby preventing them from being transformed to disease population. In ¹⁰, major categories of infectious agents and common vectors and vehicles of disease were given while Harvard health publishing in ⁹, gave some ways of preventing infections, World health Organization in ¹¹ stated that there is an urgent need to re-establish basic infection control measures and in ¹² gave measures of controlling the spread of infectious diseases, which is still very relevant today. Different authors have derived several differential operators. Author in ⁴, derived a new multiplier differential operator, in an attempt to get a class of analytic functions with finer coefficients bounds which give a better result in an application mentioned above.

For the function f of the form (1) in A, the following results are well known: f is said to be starlike respectively convex if and only if

And

Swamy ⁷, introduced a multiplier differential operator of the form defined by:

which is analytic and univalent in the unit disk. For details see ⁷.

Author in ⁵ defined a linear operator by:

(2)

and the class by:

Where

And

(3)

Now, let

(4)

We define by:

(5)

and the class by:

(6)

Where

Furthermore, let

and

we define the convolution of and by:

(7)

In this paper, we obtain the coefficients bound for the class and apply it to disease control measure.

Lemma ³: Let be as in (3). Then is in the class if and only if

Now we state and prove the main results of this paper.

2. Main Results

Theorem 1: Let be as in (5). Then belong to the class if

Proof: Given that

And

with simple calculation, we have

Where

For we have

This is bounded by if

Fixing the value of x and restructuring, we have the result.

Remark: The above theorem shows the relationship between the convexity of the integral operator in (5) and the starlikeness of the differential operator in (4).

Corollary 1: Let be as defined in (5) and belongs to the class Also, let belongs to the class as given in lemma 1. Then

Proof: Let be as defined in (5) and belongs to the class From theorem 1, we have for

For

This proves the result.

Corollary 2: Let belongs to the class Then

and

Corollary 3: Let belongs to the class . Then

Corollary 4: Let belongs to the class Then

Corollary 5: Let belongs to the class Then

Corollary 6: Let belongs to the class Then

Corollary 7: Let belongs to the class Then

Theorem 2: Let be as in (5) and Then (z) belong to the class if

Where and

Proof: Let belong to the class Then from Theorem 1, and for real value of z we have:

And

We need to find the smallest such that

(8)

By Cauchy Schwartz inequality, we have:

(9)

Thus, it suffices to show that:

From where we have:

And by (10), we have:

And by simple simplification, with and we have:

(10)

This proves the result.

Corollary 8: Let be as in (5) and belong to the class Then

and

Where is as defined in (10) and respectively.

Corollary 9: Let be as in (5) and Then

Where is as defined in (10) and respectively.

Corollary 10: Let be as in (5) and Then

Where is as defined in (10) and respectively.

In what follows, we show that the class is close under convex combination.

Theorem 3: Let be as in (5) and

Then given by

belongs to the class Where

Proof: Let and belong to the class Then we have

(11)

And respectively

(12)

adding (11) and (12) gives

This proves the result.

Now, we establish two of the fundamental theorems about univalent functions in relation to function in the subclass the growth and distortion theorems, which provide bounds on and respectively. Theorems (4) and (5) below are the growth and distortion theorems respectively.

Theorem 4: Let belong to the class Then

Proof: For the function

Similarly,

Fixing the value of for the function in and rearranging gives the result.

Theorem 5: Let belong to the class Then

Proof: The proof follows from theorem 4.

3. Conclusion

In this article, we take the coefficients of the analytic functions in the above theorems as the size of some dynamic parameters, which could be infectious agents, such as viruses, bacteria, fungi, protozoa, and helminthes, that need to be curtailed to achieve the above aim and the and are infection control measures. Thus, to achieve reduction in the infectious agents, we need to reduce the slope of the class of functions

which is the coefficients It is noted from the above that the higher the control measures, the smaller the coefficient bounds. And that the coefficient bounds in the theorems above is of less magnitude than those obtained in the previous literature, thus it will yield a better result in the prevention we aimed at.

References